It's not as complicated as some might make it sound.
If you have ever been curious about the Tarot, you may have done a bit of reading up, but been put off by all the seeming rules. Do this! Don't do that! Always do it this way! Never ever ever do this!
The truth is, handling the Tarot is very much up to interpretation. The only real rule is, "Do what feels right." Unfortunately a lot of people take that rule and run with it, and mistake "what feels right for me" with "what everyone should do all the time, lest DISASTER BEFALL THEM."
As with so many things in life, your intentions matter more than the specifics of your actions. Treat your cards with respect, approach them with an open mind and an open heart, and you will do fine.
Buying Tarot Cards
The first myth many people encounter is that "You can't buy a deck of Tarot cards for yourself, they have to be given to you as a gift." This, quite simply, is wrong. There is no reason not to buy a deck of Tarot cards for yourself. Professional diviners and Tarot card readers almost universally believe that this myth is ridiculous.
There are hundreds of different Tarot decks. The Rider-Waite deck is the most popular, but don't let that sway your purchase. You should buy whatever design speaks to you.
Don't be afraid to buy a second-hand deck. Assuming they are in good shape, used decks can be a great deal for people on a budget.
Cleansing Tarot Cards
Whether you buy your cards new or used, the next step will be cleansing them. Quite frankly, "cleansing" is more about you than the cards. Your cleansing ritual only needs to be as elaborate as you need to feel that the deck is "clean" and ready for your use.
This page has a great run-down of rituals that you can use individually or in combination. I also like their recommendation to cleanse your cards by "holding the Tarot deck firmly in your hand and striking down sharply on a table or other solid object." This gets to the heart of the ritual, which is that your intention is what manifests the change.
Storing Tarot Cards
Many experts recommend storing your cards inside a natural substance, whether it is a wooden box or a cloth made of natural material. Once again, this is up to you. If you have a beautiful swatch of cloth, feel free to use it, even if it contains man-made fibers.
As long as you are protecting your cards from physical damage, the details matter less than your feelings. You could even simply rubber band your cards together and store them in a drawer, if you did it with respect.